PhD dissertations that solve an established open problem

I find George Dantzig's story particularly impressive and inspiring.

While he was a graduate student at UC Berkeley, near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.

Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics. Neyman told Dantzig to wrap the two problems in a binder and he would accept them as a Ph.D. thesis.

The two problems that Dantzig solved were eventually published in: On the Non-Existence of Tests of "Student's" Hypothesis Having Power Functions Independent of σ (1940) and in On the Fundamental Lemma of Neyman and Pearson (1951).

Godel's Completeness Theorem, was part of his PHD thesis.

It was definitely an active field of research, but I don't know to what degree the problem was an open one, in the way we understand it today.

.. when Kurt Gödel joined the University of Vienna in 1924, he took up theoretical physics as his major. Sometime before this, he had read Goethe’s theory of colors and became interest in the subject. At the same time, he attended classes on mathematics and philosophy as well. Soon he came in contact with great mathematicians and in 1926, influenced by number theorist Philipp Furtwängler, he decided to change his subject and take up mathematics. Besides that, he was highly influenced by Karl Menger’s course in dimension theory and attended Heinrich Gomperz’s course in the history of philosophy.

Also in 1926, he entered the Vienna Circle, a group of positivist philosophers formed around Moritz Schlick, and until 1928, attended their meetings regularly. After graduation, he started working for his doctoral degree under Hans Hahn. His dissertation was on the problem of completeness.

In the summer of 1929, Gödel submitted his dissertation, titled ‘Über die Vollständigkeit des Logikkalküls’ (On the Completeness of the Calculus of Logic). Subsequently in February 1930, he received his doctorate in mathematics from the University of Vienna. Sometime now, he also became an Austrian citizen.

I am quite surprised that nobody has mentioned Grothendieck's thesis. Apparently Laurent Schwartz gave Grothendieck a recent paper listing a number of open problems in functional analysis at one of their initial meetings. (Schwartz had just won the Fields at the time.) Grothendieck went away for a few weeks/ months and then returned with solutions to many (or all?) of the questions. In the course of the next few years Grothendieck became one of the world's leading functional analysts, before turning his attention to algebraic geometry.

This is the story I heard as part of mathematical gossip many, many years ago. Maybe someone who is more knowledgeable can chime in.

Another utterly spectacular thesis was Noam Elkies'. Among other things he settled a 200 year old problem posed by Euler!